On the hyperbolicity of certain complements (Q1604960)

The authors give a new proof of the hyperbolicity of two complements of hypersurfaces in complex projective spaces. There has been a conjecture due to S. Kobayashi as follows [cf. \textit{G. Dethloff}, \textit{G. Schumacher}, and \textit{P. M. Wong}, Am. J. Math. 117, No. 3, 573-599 (1995; Zbl 0842.32021)]. Let \(S\) be a hypersurface with \(p\) components in \(\mathbb{P}_n (\mathbb{C})\). If \(S\) is generic and the degree of \(S\) is not less than \(2n+1\), then \(\mathbb{P}_n (\mathbb{C}) \setminus S\) is complete hyperbolic. For this conjecture, the following classical result due to Bloch-Green-Fujimoto is well-known [cf. \textit{M. Green}, Proc. Am. Math. Soc. 66, 109-113 (1977; Zbl 0366.32013)]: The complement of \(2n+1\) hyperplanes in general position in \(\mathbb{P}_n (\mathbb{C})\) is hyperbolic. On the other hand, \textit{H. Grauert} [Math. Z. 200, No. 2, 149-168 (1989; Zbl 0664.32020)] and \textit{G. Dethloff}, \textit{G. Schumacher} and \textit{P. M. Wong} [loc. cit. and Duke Math. J. 78, No. 1, 193-212 (1995; Zbl 0847.32028)] deal with the non hyperplane case. They prove that the complement of three generic curves in \(\mathbb{P}_2(\mathbb{C})\) is complete hyperbolic. The methods in their proofs are not elementary. In this paper the authors give proofs of the above theorems by making use of the idea of Ros [cf. \textit{L. Zalcman}, Bull. Am. Math. Soc., New. Ser. 35, No. 3, 215-230 (1998; Zbl 1037.30021)]. The proofs given in this paper are direct and elementary.